Related papers: On the average Steiner 3-eccentricity of trees

Motivated by the study of the crossing number of graphs, it is shown that, for trees, the sum of the products of the degrees of the end-vertices of all edges has an upper bound in terms of the sum of all vertex degrees to the power of…

Tree spanners approximate distances within graphs; a subtree of a graph is a tree $t$-spanner of the graph if and only if for every pair of vertices their distance in the subtree is at most $t$ times their distance in the graph. When a…

We consider the $k$-prize-collecting Steiner tree problem. An instance is composed of an integer $k$ and a graph $G$ with costs on edges and penalties on vertices. The objective is to find a tree spanning at least $k$ vertices which…

For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the…

In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral…

Let $G$ be a connected graph of order $n$. A spanning $k$-tree of $G$ is a spanning tree with the maximum degree at most $k$, and a spanning $k$-ended-tree of $G$ is a spanning tree at most $k$ leaves, where $k\geq2$ is an integer. This…

A spanning tree $T$ in a graph $G$ is a sub-graph of $G$ with the same vertex set as $G$ which is a tree. In 1981, McKay proved an asymptotic result regarding the number of spanning trees in random $k$-regular graphs. In this paper we prove…

We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\over 4}(s-2)+2$ leaves. Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively…

The eccentric connectivity index of a graph $G$ is $\xi^c(G) = \sum_{v \in V(G)}\varepsilon(v)\deg(v)$, and the eccentric distance sum is $\xi^d(G) = \sum_{v \in V(G)}\varepsilon(v)D(v)$, where $\varepsilon(v)$ is the eccentricity of $v$,…

The decycling number of a graph $G$ is the minimum number of vertices whose removal from $G$ results in an acyclic subgraph. It is known that determining the decycling number of a graph $G$ is equivalent to finding the maximum induced…

A tree $t$-spanner of a graph $G$ is a spanning tree of $G$ such that the distance between pairs of vertices in the tree is at most $t$ times their distance in $G$. Deciding tree $t$-spanner admissible graphs has been proved to be tractable…

Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We…

Let $F(G)$ be the number of forests of a graph $G$. Similarly let $C(G)$ be the number of connected spanning subgraphs of a connected graph $G$. We bound $F(G)$ and $C(G)$ for regular graphs and for graphs with fixed average degree. Among…

We study two problems that seek a subtree $T$ of a graph $G=(V,E)$ such that $T$ satisfies a certain property and has minimal maximum degree. - In the Min-Degree Group Steiner Tree problem we are given a collection ${\cal S}$ of groups…

A $k$-nearly independent vertex subset of a graph $G$ is a set of vertices that induces a subgraph containing exactly $k$ edges. For $k = 0$, this coincides with the classical notion of independent subsets. This paper investigates the…

Two classical concepts of centrality in a graph are the median and the center. The connected notions of the status and the radius of a graph seem to be in no relation. In this paper, however, we show a clear connection of both concepts, as…

The Steiner Tree problem is a classical problem in combinatorial optimization: the goal is to connect a set $T$ of terminals in a graph $G$ by a tree of minimum size. Karpinski and Zelikovsky (1996) studied the $\delta$-dense version of…

The \emph{eccentricity} of a vertex $u$ in a graph $G$, denoted by $e_G(u)$, is the maximum distance from $u$ to other vertices in $G$. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity…

Tree-decompositions and treewidth are of fundamental importance in structural and algorithmic graph theory. The "spread" of a tree-decomposition is the minimum integer $s$ such that every vertex lies in at most $s$ bags. A…

Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor +m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. In this paper, we give a…